[drdeca]

I'm familiar with the cantor set, and I know it has 0 measure. Just because you can succinctly describe the cantor set, which has 0 measure, doesn't mean I've messed up. If I assign a uniform distribution over [0,1] to some number outcome in the world, and an element in the cantor set is the result, then I've messed up. But, when we measure numbers in the world, we don't measure specific real numbers, as all our measurements have some amount of error. So, that can't happen. We can measure that the result is in some interval, and that this interval contains some element of the cantor set, but the probability of what we observed, is not something that I assigned 0 probability to. Like, heck, every interval will have a rational number in it, and the rational numbers also have measure 0.

"the measured value is in the cantor set" isn't a thing that we can observe to have happened.

("the value, when rounded to the finite amount of precision that our measurement has, is in the cantor set" is something that would have positive probability, under the uniform distribution over the interval, and therefore something I shouldn't assign a probability of 0.)